Download Evolving technical trading rules for spot foreign

CMS 1: 311–327 (2004)
DOI: 10.1007/s10287-004-0018-5
Evolving technical trading rules
for spot foreign-exchange markets
using grammatical evolution
Anthony Brabazon1 , Michael O’Neill2
1 University College Dublin, Ireland (e-mail: [email protected])
2 University of Limerick, Ireland (e-mail: [email protected])
Abstract. Grammatical Evolution (GE) is a novel, data-driven, model-induction
tool, inspired by the biological gene-to-protein mapping process. This study provides an introduction to GE, and applies the methodology in an attempt to uncover
useful technical trading rules which can be used to trade foreign exchange markets.
In this study, each of the evolved rules (programs) represents a market trading system. The form of these programs is not specified ex-ante, but emerges by means
of an evolutionary process. Daily US-DM, US-Stg and US-Yen exchange rates for
the period 1992 to 1997 are used to train and test the model. The findings suggest that the developed rules earn positive returns in hold-out sample test periods,
after allowing for trading and slippage costs. This suggests potential for future
research to determine whether further refinement of the methodology adopted in
this study could improve the returns earned by the developed rules. It is also noted
that this novel methodology has general utility for rule-induction, and data mining
applications.
Keywords: Grammatical evolution, Foreign exchange prediction, Technical trading rules
MSC codes: 91B84, 68W10, 62J02
1 Introduction
The last decade has seen significant advances in the field of computational intelligence, leading to the development of powerful new modelling technologies.
Generally, these technologies fall into three categories, those which are inspired by
the workings of biological neurons (Neural Networks), those which are inspired by
CMS Computational Management Science
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an evolutionary metaphor (Genetic Algorithms, Genetic Programming and Grammatical Evolution), and those which are inspired by studies of animal interactions
(Particle Swarm and Ant Colony models). While neural networks, and to a lesser extent genetic algorithms and ant-algorithms, have featured in Management Science
applications, other forms of computational intelligence have received relatively less
attention.
Grammatical Evolution (GE), represents an evolutionary automatic programming
methodology, and can be used to evolve ‘rule sets’. These rule sets can be as general
as a functional expression which produces a good mapping between a series of
known input-output data vectors. A particular strength of the methodology is that
the form of the model need not be specified a priori by the modeller. This is of
particular utility in cases where the modeller has a theoretical or intuitive idea of
the nature of the explanatory variables, but a weak understanding of the functional
relationship between the explanatory and the dependent variable(s). GE does not
require that the model form is linear, nor does the method require that the measure
of model error used in model construction is a continuous or differentiable function.
A key element of the methodology is the concept of a Grammar, which governs the
creation of the rule sets. The concept of a Grammar is inspired by the biological
process of mapping of genes to proteins. This paper introduces the GE methodology
to Management Science, and applies the methodology in an attempt to uncover a
series of useful technical trading rules for trading in spot foreign exchange markets
(transactions in which two currencies are immediately exchanged).
1.1 Foreign exchange markets
Foreign exchange markets are the most active of all financial markets with average
daily trading volumes in traditional (non-electronic broker) foreign exchange markets estimated at $1.2 trillion (Bank of International Settlements, 2001). The average
daily trading volume on spot markets exceeds $387 billion. Although the precise
scale of speculative trading on spot markets is unknown, the bulk of transactions
are not represented by world trade in goods and services (Froot and Thaler, 1990).
Only about 15% of the trading is driven by non-dealer/financial institution trading
(Bank of International Settlements, 2001). Therefore, it is plausible to assume that
the scale of speculative trading is substantial. Speculative foreign currency trading
implies the existence of predictive models in the mind of investors. These models could integrate many different forms of information. This study focuses on a
subset of this information, that contained in the historical time-series of exchange
rates. International foreign-exchange markets are dominated by a small number
of heavily-traded currency pairings. Approximately 90% of all foreign currency
transactions involve the US Dollar (Bank of International Settlements, 2001). Prior
to the recent introduction of the Euro, the most-traded currency pairings were the
US Dollar-DM, US Dollar-Yen and US Dollar-Sterling, accounting for 20%, 18%
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and 8% of total daily trading volume in 1998 (Bank of International Settlements,
2001). This study develops trading systems for each of these currency pairs.
1.2 Technical analysis
Technical analysis is defined as the attempt to identify regularities in the time-series
of price and volume information from a financial market, by extracting patterns from
noisy data (Lo, Mamaysky and Wang, 2000). Some market traders known as technical analysts, believe that prices (exchange rates) move in trends and that price
patterns repeat themselves (Murphy, 1999). Although controversy exists amongst
financial theorists regarding the veracity of the claims of technical analysts, the
methods are widely applied in practice. In a study (Taylor and Allen, 1992) conducted on behalf of the Bank of England, it was found that approximately 90% of
financial institutions dealing in foreign exchange in London placed some weight
on information obtained from technical analysis. Greatest use was made of this
information in forming short-run (a trading horizon of less than three months) exchange rate expectations. In the case of foreign-exchange markets, studies which
suggest that technical trading strategies may be profitable include Sweeney (1986)
and Levich and Thomas (1993). Further support for a technical analysis argument
is found in Osler (2003), where evidence of clustering of currency stop-loss and
take-profit orders is noted. The affect of clustering of these orders is that trends
in exchange rates will tend to reverse at predictable support and resistance levels,
but trends tend to accelerate once these levels are breached. Clusters of take-profit
orders were found at exchange rates characterized by ‘round numbers’ (ending
in 00) with stop-loss orders clustering just beyond round numbers. If we accept
the premise that there are rules, although not necessarily static rules, which partly
explain price behavior it follows that opportunities may exist to enhance trading
decisions through use of an appropriate rule induction methodology such as GE.
1.3 Technical Indicators
Technical indicators are formed from various combinations of current and historic
price (volume) information. Four groupings of indicators are given prominence in
the technical analysis literature (Brock, Lakonishok and LeBaron, 1992; Murphy,
1999; Pring, 1991):
i.
ii.
iii.
iv.
Moving average indicators
Momentum indicators
Trading range indicators
Oscillators
In a study of the application of technical analysis in the foreign exchange markets, Taylor and Allen (1992) found that the most common indicators utilized were
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moving averages, followed by momentum and oscillators. These findings are supported by the recent study of Eun and Sabherwal (2002) which provides empirical
evidence, based on currency rate forecasts of major banks, that banks tend to be
momentum forecasters. Given the large search space, an evolutionary automatic
programming methodology has promise to determine both a good quality combination of, and relevant parameters for, trading rules drawn from individual technical
indicators. In this paper, we have limited our attention to the first three groupings
of indicators.
The simplest moving average systems compare the current exchange rate with a
moving average of the exchange rate over a lagged period, to determine how far
the current rate has moved from an underlying trend. As they smooth out daily
price fluctuations, moving averages can heighten the visibility of an underlying
trend. A trading signal can be generated when an exchange rate moves above or
below its moving average. A variation on simple moving average systems is to use
a moving average convergence-divergence (MACD) oscillator. This is calculated
by taking the difference of a short run and a long run moving average. In a recursive
fashion, more complex combinations of moving averages of values calculated from
a MACD oscillator can themselves be used to generate trading rules. The momentum
of an exchange rate is the ratio of a time-lagged exchange rate to the current rate
(Ratet /Ratet−x ). The belief underlying this indicator is that a strong tend is likely
to persist for a period of time. Trading Range Breakout indicators seek to determine
when the price of a currency moves out of a range defined by its maximum or
minimum value in a lagged time-period. A simple example of a trading rule derived
from a breakout indicator would be to buy a currency when it exceeds its previous
high (against another currency) in the last four weeks and conversely to sell if it
falls below its previous four week low.
1.4 Motivation for study
Although many studies have examined the potential for technical trading since the
move to a floating exchange rate system in the 1970s, typically studies employ a
limited, fixed, sub-set of the population of technical indicators. In most studies,
the lag periods of the indicators are fixed and the modelling task is the selection
of a subset of the pre-specified indicators along with their coefficients in order to
minimize model-fit error. As noted by Iba and Nikolaev (2000) there are a number
of reasons to suppose that the use of an evolutionary automatic programming (EAP)
approach, such as GE, can prove fruitful in the financial prediction domain. EAP
can conduct an efficient exploration of a search space and can uncover dependencies
between input variables, leading to the selection of a good subset for inclusion in
the final model. Additionally, use of EAP facilitates the implementation of complex
fitness functions including discontinuous, non-differentiable functions. This is of
particular importance in a financial domain as fitness criterion may be complex,
usually requiring a balancing of risk and return.
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Although quite a number of studies have applied artificial neural networks to
the problem of foreign exchange rate prediction (Yao and Tan, 2000; Hu, Zhang,
Jiang and Patuwo, 1999; Jamal and Sundar, 1999; Gencay, Dacorogna, Olsen and
Pictet, 2003; Franses and Van Homelen, 1998), there have been fewer applications
of Genetic Algorithm (GA) (Holland, 1975; Goldberg, 1989) or Genetic Programming (GP) (Koza, 1992; Banzhaf et al., 1998) methodologies to the prediction of
financial markets using technical indicators. Notable exceptions include (Allen and
Karjalainen, 1999; Colin, 1994; Bauer, 1994; Neely, Weller and Dittmar, 1997;
Deboeck, 1994; O’Neill, Brabazon and Ryan, 2002) which concentrated on equity
markets. However, little published work other than Bhattacharyya, Pictet, Zumbach
(2002) exists concerning the application of evolutionary-inspired algorithms to the
technical trading of foreign exchange markets. The need to research the potential
of EAP methodologies in the foreign exchange markets has been recognized in the
recent past (Jegadeesh, 2000). This study answers this call, and introduces GE to
the problem domain.
The rest of this paper is organized as follows. The next section provides an introduction to Grammatical Evolution. Then we outline the data set and methodology
utilized. The following sections provide the results of the study followed by a
number of conclusions.
2 Grammatical evolution
A full description of GE can be found in (O’Neill and Ryan, 2003; O’Neill and
Ryan, 2001; O’Neill, 2001; Ryan, Collins and O’Neill, 1998). GE is an evolutionary algorithm that can evolve computer programs in any language and as such is
an example of an evolutionary automatic programming system. We use the term
evolutionary automatic programming to refer to those systems that adopt evolutionary computation to automatically generate computer programs and as such it
encompasses Genetic Programming (GP) (Koza, 1992; Koza, 1994; Banzhaf et al.,
1998; Koza et al., 1999; Koza et al., 2003) and all its variants. For a full overview
of the background on GE and its context in evolutionary automatic programming,
the reader is referred to (O’Neill and Ryan, 2003). Rather than representing the
programs as syntax trees, as in GP, a linear genome representation is used. Each individual, a variable length binary string, contains in its codons (groups of 8 bits) the
information to select production rules from a Backus Naur Form (BNF) grammar.
In other words, an individuals binary string contains the instructions that direct a
developmental process resulting in the creation of a program or expression. As such,
GE adopts a biologically-inspired, genotype-phenotype mapping process. Future
advances in the field of evolutionary algorithms can be easily incorporated into this
system due to the separation of phenotype from genotype, as the mapping process
at the heart of GE can be plugged onto any search algorithm that can operate over
binary or integer vectors.
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Grammatical Evolution
Biological System
Binary String
DNA
TRANSCRIPTION
Integer String
RNA
TRANSLATION
Rules
Program /
Function
Executed Program
Amino
Acids
Protein
Phenotypic Effect
Fig. 1. A comparison between the grammatical evolution system and a biological genetic system. The
binary string of GE is analogous to the double helix of DNA, each guiding the formation of the phenotype.
In the case of GE, this occurs via the application of production rules to generate the terminals of the
compilable program. In the biological case by directing the formation of the phenotypic protein by
determining th e order and type of protein subcomponents (amino acids) that are joined together
2.1 The biological approach
The GE system is inspired by the biological process of generating a protein from the
genetic material of an organism. Proteins are fundamental in the proper development
and operation of living organisms and are responsible for traits such as eye colour
and height (Lewin, 2000).
The genetic material (usually DNA) contains the information required to produce
specific proteins at different points along the molecule. For simplicity, consider
DNA to be a string of building blocks called nucleotides, of which there are four,
named A, T, G, and C, for adenine, tyrosine, guanine, and cytosine respectively.
Groups of three nucleotides, called codons, are used to specify the building blocks of
proteins. These protein building blocks are known as amino acids, and the sequence
of these amino acids in a protein is determined by the sequence of codons on the
DNA strand. The sequence of amino acids is very important as it plays a large part
in determining the final three-dimensional structure of the protein, which in turn
has a role to play in determining its functional properties.
In order to generate a protein from the sequence of nucleotides in the DNA, the
nucleotide sequence is first transcribed into a slightly different format, that being a
sequence of elements on a molecule known as mRNA. Codons within the mRNA
molecule are then translated to determine the sequence of amino acids that are
contained within the protein molecule. The application of production rules to the
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non-terminals of the incomplete code being mapped in GE is analogous to the role
amino acids play when being combined together to transform the growing protein
molecule into its final functional three-dimensional form.
The result of the expression of the genetic material as proteins in conjunction with
environmental factors is the phenotype. In GE, the phenotype is a computer program
that is generated from the genetic material (the genotype) by a process termed a
genotype-phenotype mapping. This is unlike the standard method of generating a
solution (a program in the case of GE) directly from an individual in an evolutionary
algorithm by explicitly encoding the solution within the genetic material. Instead,
a many-to-one mapping process is employed within which the robustness of the
GE system lies. Figure 1 compares the mapping process employed in both GE and
biological organisms.
2.2 The mapping process
When tackling a problem with GE, a suitable BNF (Backus Naur Form) grammar
definition must first be decided upon. The BNF can be either the specification of an
entire language or, perhaps more usefully, a subset of a language geared towards
the problem at hand.
In GE, a BNF definition is used to describe the output language to be produced by
the system. BNF is a notation for expressing the grammar of a language in the form
of production rules. BNF grammars consist of terminals, which are items that
can appear in the language, e.g. binary operators +, -, unary operators Sin, constants
1.0 etc. and non-terminals, which can be expanded into one or more terminals and non-terminals. For example from the grammar detailed below, <expr>
can be transformed into one of four rules, i.e it becomes <expr><op><expr>,
(<expr><op><expr>) (which is the same as the first, but surrounded by brackets), <pre-op>(<expr>), or <var>.A grammar can be represented by the tuple
{N, T , P , S}, where N is the set of non-terminals, T the set of terminals, P a set of
production rules that maps the elements of N to T , and S is a start symbol which is
a member of N . When there are a number of production rules that can be applied
to one element of N the choice is delimited with the ‘|’ symbol. For example,
Table 1. The number of choices available from each production rule
Rule no.
Choices
A
B
C
D
4
4
1
2
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N = { <expr>, <op>, <pre_op> }
T = {Sin, +, -, /, *, X, 1.0, (, )}
S = <expr>
And P can be represented as:
(A) <expr> ::=
|
|
|
(B) <op> ::=
|
|
|
<expr> <op> <expr>
(0)
( <expr> <op> <expr> ) (1)
<pre-op> ( <expr> )
(2)
<var>
(3)
+
/
*
(0)
(1)
(2)
(3)
(C) <pre-op> ::= Sin
(D) <var> ::= X
| 1.0
(0)
(1)
The compilable code produced will consist of elements of the terminal set T . The
grammar is used in a developmental approach whereby the evolutionary process
evolves the production rules to be applied at each stage of a mapping process,
starting from the start symbol, until a complete program is formed. A complete
program is one that is comprised solely from elements of T .
As the BNF definition is a plug-in component of the system, it means that GE can
produce code in any language thereby giving the system a unique flexibility. For
the above BNF, Table 1 summarizes the production rules and the number of choices
associated with each.
The genotype is used to map the start symbol onto terminals by reading codons
of 8 bits to generate a corresponding integer value, from which an appropriate
production rule is selected by using the following mapping function:
Rule = Codon V alue % N o. Rule Choices
Consider the following rule from the given grammar. Given the non-terminal op,
there are four production rules which can be selected which will replace op with a
terminal.
(B) <op> :: =
+
| | /
| *
(0)
(1)
(2)
(3)
If we assume the codon being read produces the integer 6, then
6%4 = 2
would select rule (2) /. Each time a production rule has to be selected to transform
a non-terminal, another codon is read. In this way the system traverses the genome.
During the genotype-to-phenotype mapping process, it is possible for individuals to run out of codons, and in this case we wrap the individual and reuse the
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319
codons. This is quite an unusual approach in EAs, as it is entirely possible for certain
codons to be used two or more times. This technique of wrapping the individual
draws inspiration from the gene-overlapping phenomenon that has been observed
in many organisms (Lewin, 2000).
In GE, each time the same codon is expressed it will always generate the same
integer value, but, depending on the current non-terminal to which it is being applied, it may result in the selection of a different production rule. We refer to this
feature as intrinsic polymorphism. What is crucial, however, is that each time a particular individual is mapped from its genotype to its phenotype, the same output is
generated. This is the case because the same choices are made each time. However,
it is possible that an incomplete mapping could occur, even after several wrapping
events, and in this case the individual in question is given the lowest possible fitness value. The selection and replacement mechanisms then operate accordingly to
increase the likelihood that this individual is removed from the population.
An incomplete mapping could arise if the integer values expressed by the genotype were applying the same production rules repeatedly. For example, consider an
individual with three codons, all of which specify rule 0 from below:
(A) <expr> :: =
<expr><op><expr>
|(<expr><op><expr>)
|<pre-op>(<expr>)
|<var>
(0)
(1)
(2)
(3)
even after wrapping the mapping process would be incomplete and would carry on
indefinitely unless stopped. This occurs because the nonterminal <expr> is being
mapped recursively by production rule 0, becoming
<expr><op><expr>. Therefore, the leftmost <expr> after each application of
a production would itself be mapped to a <expr><op><expr>, resulting in a
continually growing expression. Such an individual is dubbed invalid as it will never
undergo a complete mapping to a set of terminals. For this reason we impose an
upper limit on the number of wrapping events that can occur. It is clearly essential
that stop sequences are found during the evolutionary search in order to complete the
mapping process to a functional program. The stop sequence being a set of codons
that result in the non-terminals being transformed into elements of the grammars
terminal set.
Beginning from the left hand side of the genome then, codon integer values are
generated and used to select rules from the BNF grammar, until one of the following
situations arise:
i. A complete program is generated. This occurs when all the non-terminals in
the expression being mapped are transformed into elements from the terminal
set of the BNF grammar.
ii. The end of the genome is reached, in which case the wrapping operator is
invoked. This results in the return of the genome reading frame to the left hand
side of the genome once again. The reading of codons will then continue, unless
an upper threshold representing the maximum number of wrapping events has
occurred during this individual’s mapping process.
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iii. In the event that a threshold on the number of wrapping events has occurred
and the individual is still incompletely mapped, the mapping process is halted,
and the individual is assigned the lowest possible fitness value.
To reduce the number of invalid individuals being passed from generation to
generation, a steady state replacement mechanism is employed. One consequence
of the use of a steady state method is its tendency to maintain fit individuals at the
expense of less fit, and in particular, invalid individuals.
In this study, the GE algorithm uses a steady state replacement mechanism,
such that, two parents produce two children the best of which replaces the worst
individual in the current population, if the child has greater fitness. The standard
genetic operators of bit mutation (probability of 0.01), and crossover (probability
of 0.9) are adopted as is a duplication operator (probability of 0.01), that duplicates a random codon and inserts this into the penultimate codon position on the
genome. A series of functions, in this study, technical indicators, are pre-defined as
are a series of mathematical operators. A population of initial trading rulesets (programs) are randomly generated, and by means of an evolutionary process, these are
improved. No explicit model specification is assumed ex-ante, although the choice
of mathematical operators defined in the grammar do place implicit limitations on
the model specifications amongst which GE can search.
3 Experimental approach
This study uses daily closing exchange rate data drawn from the London market
for the period 23/10/92 to 13/10/97. The training data set was comprised of the
first 799 trading days of the data set. In the application of any iterative modelling
technique it is important to consider the danger of data-snooping. Data-snooping
can occur when a dataset is used more than once in a model construction process
(Sullivan, Timmermann and White, 1999). To reduce this danger, the developed
models are tested using an extensive out-of-sample dataset of 548 trading days.
The out-of-sample data is divided into two hold-out samples (274 * 2) to allow
comparison of the hold-out results across different market conditions. This is done
in order to assess the stability and degradation characteristics of the developed
models’ predictions. The rules evolved by GE are used to generate one of three
signals for each day of the training or test periods. The possible signals are Buy,
Sell, or Do-Nothing. Permitting the model to output a Do-Nothing signal reduces the
hard threshold problem associated with production of a binary output. Markets do
not invariably trend upwards or downwards, but may trade sideways in a relatively
narrow band of prices (a trading range). Trend-following trading systems will tend
to struggle in sideways-moving markets, and hence should be allowed to ‘opt-out’
of trading these markets for which they are not well-suited.
Evolving technical trading rules for spot foreign-exchange markets
321
3.1 Trading methodology
A variant on the trading methodology developed in Brock, Lakonishok and LeBaron
(1992) is then applied. If a buy signal is indicated on a given day, a fixed investment
of $1,000 is made in the foreign currency using borrowed funds. This position is
automatically closed at the end of a five day trading period. On the production of a
sell signal, an investment of $1,000 is made on the short side and again this position
is closed out after a five day period. This gives rise to a maximum potential investment of $5,000 at any point in time.1 The total return (measured as a percentage)
generated by the developed trading system is a combination of its trading return
net of trading costs and an interest differential.
The technical indicators adopted in this study are the moving average, momentum and trading range breakout where the relevant periods and combination
of indicators used are to be determined by evolution. Apart from the raw components of the technical indicators adopted (average, moving average, lag, each of
which takes a real-valued argument representing the chosen time-window) the BNF
grammar also allows the use of standard arithmetic operators (+, -, * and inequality
operators) and the binary operators f_and, f_or, and the unary operator f_not.2 The
current day’s price is also provided as a model input. The signals generated for each
day are post-processed using the following rule:
Sell = V alue < .33
Do − N othing = .33 >= V alue < .66
Buy = .66 >= V alue
Consider a daily observed exchange rate series St with t = 1, 2, ...n. We define
t
the five day (%) trading return (before trading costs) for a long position as ( SSt+5
−1)
and as −1[1 − ( SSt+5
)] for a short position.
t
In this trading strategy, the home currency is the US dollar. When a long position
is taken, the foreign currency is purchased and the return equals the trading return
plus the interest earned in the foreign currency, less the cost of the home funds
borrowed to invest in the foreign currency. When a short signal is produced by the
trading system, the foreign currency is borrowed, at a cost of the foreign interest
rate, and is invested in US dollars for the trading period. The dollar interest rate
is earned on these funds. A trading cost c of 0.025% and a slippage allowance of
0.01% in each (single) direction is included (Levich and Thomas, 1993).
To encourage the evolution of a trading system with good risk to return characteristics, we penalize trading systems leading to volatile patterns of returns. The
1 The potential loss on an unprotected short sale is in theory infinite but in practice is unlikely to
exceed $1,000. On a short sale, an amount of $1,000 is considered invested for the five day period of
the transaction.
2 The operations f_and(argument 1, argument 2), f_or (argument 1, argument 2), and f_not (argument
1) return the minimum, maximum, of the arguments, and 1 - the argument, respectively.
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risk of catastrophic loss is reduced by incorporating a measure of the system’s
drawdown (maximum cumulative loss since commencing trading with the system)
into the fitness function:
F itness = Return − x ∗ (Max.drawdown)
‘x’ can be considered as a tuning parameter for the penalty function. The more riskadverse the investor, in terms of wishing to avoid a catastrophic loss, the higher the
value of this parameter. We initially set this parameter to 1.
3.2 Benchmark for trading system
Before the returns generated by the trading system can be assessed, an appropriate benchmark must be defined. A wide variety of benchmarks could be utilized
including:
i A no-change model
ii A perfect foresight model
iii Simple buy-and-hold
The no-change model assumes that foreign-exchange rates follow a randomwalk, therefore the current (spot) foreign exchange rate is the best predictor of the
future rate. Under this perspective, the expected rate of return from speculating
in foreign currencies, net of financing costs, is zero. The perfect foresight model
compares the generated trading returns with the returns the investor could have
generated if he had perfect foresight of all future changes in the exchange rate.
This is a very stringent benchmark as we know that financial time-series have a low
signal to noise ratio. This benchmark is not used in this study. A simple buy-andhold strategy results when an investor buys the foreign currency long or short, and
closes out the position at the end of the trading period. This provides a basic test of
whether the trading system is covering its associated trading costs. Although this
is a simple metric, it is not risk-comparable with the trading model, as it maintains
a fully-invested position in the currency markets at all times. To the extent that the
buy-and-hold strategy maintains a greater investment in currency markets than the
trading model, it has more capital at risk.
4 Results
The results from our experiments are now provided. 30 independent trials were
conducted with a population size of 500 individuals and an evolutionary period
of 50 generations. To assess the quality of the results obtained they are compared
against a ‘buy-and-hold’ benchmark. The ‘buy-and-hold’ benchmark is determined
by comparing the return, measured as the percentage gain on the initial investment,
obtained from investing $5,000 in the foreign currency, net of the interest rate
differential, during each training and out-of-sample period.
Evolving technical trading rules for spot foreign-exchange markets
Grammatical Evolution - ForEx - US/DM
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Grammatical Evolution - ForEx - US/Yen
Grammatical Evolution - ForEx - US/STG
2
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best
avg
best
avg
best
avg
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Fig. 2. In-sample fitness of overall best individual and mean of best individuals over the 30 runs, plotted
for each currency pair
Table 2. Results for the best (mean) evolved rule set over the 30 runs compared to the benchmark
buy-and-hold strategy on the US-DM dataset
Trading period
Evolved Rule set
Buy-and-Hold
Training
Test 1
Test 2
2.02357 (1.865)
0.017998 (0.017998)
0.302671 (0.302671)
-0.112199
0.068231
0.153231
4.1 US-DM
A plot of the mean best and mean average fitness at each generation over the 30
runs can be seen in Figure 2. Table 2 provides a comparison of the performance
(percentage return on the initial capital of $5,000, where 0.01 = 1%) of the best
evolved ruleset against that of the benchmark investment strategy.
4.2 US-STG
Table 3 provides a comparison of the performance of the best evolved ruleset against
that of the benchmark investment strategy.
4.3 US-Yen
Table 4 provides a comparison of the performance of the best evolved ruleset against
that of the benchmark investment strategy.
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A. Brabazon, M. O’Neill
Table 3. Results for the best (mean) evolved rule set over the 30 runs compared to the benchmark
buy-and-hold strategy on the US-STG dataset
Trading period
Evolved Rule set
Buy-and-Hold
Training
Test 1
Test 2
1.75642 (1.5123)
0.343709 (0.26103)
0.500649 (0.47824)
−0.055855
−0.094723
−0.041456
Table 4. Results for the best (mean) evolved rule set over the 30 runs compared to the benchmark
buy-and-hold strategy on the US-Yen dataset
Trading period
Evolved Rule set
Buy-and-Hold
Training
Test 1
Test 2
2.3803 (1.9374)
0.342609 (0.066474)
0.195778 (0.32521)
-0.080146
0.133476
0.090996
5 Discussion
A total of 799 days of data were used to train the models, which were then tested
on a further 548 days of out-of-sample data. The out-of-sample data is split into
two sub-samples of 274 days, in order to assess whether the results of the trading
system degrade noticeably during the out-of-sample period. Despite the lengthy
out-of-sample period, the evolved trading rules generated positive returns on all
the hold-out samples, after allowance for trading, slippage and net interest costs.
In five of the six hold-out-periods, the best evolved rule notably outperforms the
benchmark buy-and-hold strategy. The mean (over all 30 runs) best evolved rule
outperforms the benchmark in four of the six hold-out periods. It is also notable that
the out-of-sample results appear robust over the two out-of-sample periods (‘Test
1’ and ‘Test 2’), and do not indicate that the performance of the trading system is
declining.
In evaluating the performance of any market timing system, a number of caveats
must be borne in mind. Any trading model constructed and tested using historic data
will tend to perform less well in a live environment than in a back-test period for
a number of reasons. Live markets have attendant problems of delay in executing
trades, illiquidity, interrupted / corrupted data and interrupted markets. The impact
of these issues is to raise trading costs and consequently to reduce trading profitability. An allowance for these costs (slippage) has been included in this study but it
is impossible to determine the scale of these costs ex-ante with complete accuracy.
In addition, markets are competitive. As new computational technologies spread,
opportunities to utilize these technologies to earn excess risk-adjusted profits are
eroded (Gencay, Dacorogna, Olsen and Pictet, 2003). Hence, estimates of trading
performance based on historical data may not be replicated in live trading as other
market participants will apply similar technologies. It must also be remembered
Evolving technical trading rules for spot foreign-exchange markets
325
that foreign exchange trading takes place 24 hours per day, hence there is no true
opening or closing price for a foreign exchange market. In view of the difficulties
in constructing and testing models using continuous data over a prolonged period,
this study like most others, adopts a sampling approach by employing daily closing
exchange rate data, drawn from the London market.
6 Conclusions & Future work
In this paper a novel methodology GE was introduced, and applied for the purposes
of foreign-exchange prediction. It is noted that this novel methodology has general
utility for rule-induction, data mining applications. GE is an evolutionary algorithm
that can evolve ‘rulesets’. In this study each of the evolved rulesets represents a
market trading system. GE was shown to evolve trading rules that were superior to
the benchmark buy-and-hold strategy in five of the six hold-out periods. A number
of additional extensions of this work are left for future development. One extension
would be to incorporate a more complex model of learning (forgetting). Glassman
(1973) suggested that the “fallibility of memory” (p. 88) may represent a useful
adaptive device when faced with a dynamic environment. At present in our model,
all historic data observations are given equal weighting which implicitly assumes
model stationarity. By a suitable modification of the fitness function, whereby more
recent data observations are assigned a higher weighting in the model construction
process, model development could be biased towards more recent data (Refenes
et al., 1997). The weighting parameter could also be evolved as a component of
the developed model. Our methodology for this initial study has included a number of simplifications, for example we only considered a small set of technical
indicators, one metric for trading risk, and a relatively low frequency sampling of
foreign-exchange data. The study could also be extended by constructing a longer
horizon predictive model using GE, employing a series of fundamental as distinct
from technical indicators. Scope also exists to develop more sophisticated moneymanagement strategies than those employed in this study. For example, rather than
inventing for a set period of five trading days, each open position could be reassessed every day, with stop-losses being used to reduce the risk of major losses.
Another interesting strategy is to vary the amount invested on each trade. Rather
than investing a fixed amount in the currency market on each trade, the investment
amount could be linked to the strength of the trading signal, with strong signals
resulting in the assumption of a bigger position than weaker signals.
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